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Entropy of scalar reaction-diffusion equations. (English) Zbl 1349.37080
Summary: We consider scalar reaction-diffusion equations on bounded and extended domains, both with the autonomous and time-periodic nonlinear term. We discuss the meaning and implications of the ergodic Poincaré-Bendixson theorem to dynamics. In particular, we show that in the extended autonomous case, the space-time topological entropy is zero. Furthermore, we characterize in the extended nonautonomous case the space-time topological and metric entropies as entropies of a pair of commuting planar homeomorphisms.

MSC:
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37B40 Topological entropy
35B40 Asymptotic behavior of solutions to PDEs
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